So I have to state whether the statement is true or false then explain why it is true or give a counter-example. The statement is as follows: If $G_1 \cong G_2$ and $H_1 \cong H_2$ then $G_1 \oplus H_1 \cong G_2 \oplus H_2$. I am unsure of how to go about this problem and where to start really on it.
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$\begingroup$ Have you written down the definition of what it means for $G_1\cong G_2$ to be true, and what $G_1\oplus H_1$ means formally, down on paper side by side? $\endgroup$– Santana AftonCommented Nov 10, 2017 at 7:32
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$\begingroup$ It's good that you're careful about statements like these. Some similar-looking statements are false: math.stackexchange.com/q/1471483/55540 $\endgroup$– Henry SwansonCommented Nov 10, 2017 at 7:44
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You can write the isomorphism explicitely. First start with two isomorphisms:
$$f:G_1\to G_2$$ $$g:H_1\to H_2$$
and then define
$$h:G_1\oplus H_1\to G_2\oplus H_2$$ $$h(x, y)=\big(f(x), g(y)\big)$$
I leave it as an exercise to check that $h$ is an isomorphism.
How did I come up with the formula? Well, if you look at $f, g$ and look at domain and codomain of $h$ you will realize that there is not much choice. It's the first idea you should have, the natural one. It's just a matter of whether it works or not. And in this case it does.
